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Thread: differentiate

  1. #1
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    differentiate

    differentiate with respect to t where t is real $\displaystyle \frac{w+t}{|w+t|^2}$ where w is a constant complex number. I get $\displaystyle \frac{|z+t|^2-2(z+t)^2}{|z+t|^4}$ but that is the wrong answer in my book. Thanks
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    Re: differentiate

    Quote Originally Posted by Plato13 View Post
    differentiate with respect to t where t is real $\displaystyle \frac{w+t}{|w+t|^2}$ where w is a constant complex number. I get $\displaystyle \frac{|z+t|^2-2(z+t)^2}{|z+t|^4}$ but that is the wrong answer in my book. Thanks
    Note that $\displaystyle | w + t |^2 = (w + t)^2$ is always positive. Thus
    $\displaystyle \frac{w + t}{ |w + t| ^2 } = \frac{w + t}{(w + t)^2}$

    What can you do with this?

    -Dan
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  3. #3
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    Re: differentiate

    W is a complex number so this is not true.
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    Re: differentiate

    Quote Originally Posted by Plato13 View Post
    differentiate with respect to t where t is real $\displaystyle \frac{w+t}{|w+t|^2}$ where w is a constant complex number. I get $\displaystyle \frac{|z+t|^2-2(z+t)^2}{|z+t|^4}$ but that is the wrong answer in my book. Thanks
    Recall that $\displaystyle |z|^2=z\cdot\overline{z}$ and because $\displaystyle t$ is real then $\displaystyle \overline{t}=t.$

    Thus $\displaystyle |w+t|^2=(w+t)\overline{(w+t)}=|w|^2+2\text{Re}(w)t +t^2~$. So$\displaystyle \frac{d(|w+t|^2)}{dt}=2\text{Re}(w)+2t$
    Last edited by Plato; May 12th 2013 at 08:29 AM.
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    Re: differentiate

    Quote Originally Posted by Plato View Post
    Recall that $\displaystyle |z|^2=z\cdot\overline{z}$ and because $\displaystyle t$ is real then $\displaystyle \overline{t}=t.$

    Thus $\displaystyle |w+t|^2=(w+t)\overline{(w+t)}=|w|^2+\text{Re}(w)t+ t^2~$. So$\displaystyle \frac{d(|w+t|^2)}{dt}=\text{Re}(w)+2t$
    Plato, shouldn't that be 2Re(w)t?
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    Re: differentiate

    Quote Originally Posted by Plato13 View Post
    Plato, shouldn't that be 2Re(w)t?
    Right! I will edit.
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    Re: differentiate

    Thanks for the help
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